DIFFERENTIAL GEOMETRY: AN OVERVIEW

We must confess in all humility that, while number is a product of our mind alone, space has a reality beyond the mind whose rules we cannot completely prescribe. (1830)

Gravitation is a manifestation of spacetime curvature, and that curvature shows up in the deviation of one geodesic from a nearby geodesic ("relative acceleration of test particles"). The central issue of this part of the book is clear: How can one quantify the "separation," and the "rate of change" of "separation," of two "geodesics" in "curved" spacetime? A clear, precise answer requires new concepts.
"Separation" between geodesics will mean "vector." But the concept of vector as employed in flat Lorentz spacetime (a bilocal object: point for head and point for tail) must be sharpened up into the local concept of tangent vector, when one passes to curved spacetime. Chapter 9 does the sharpening. It also reveals how the passage to curved spacetime affects 1 -forms and tensors.

It takes one tool (vectors in curved geometry, Chapter 9) to define "separation" clearly as a vector; it takes another tool (parallel transport in curved spacetime, Chapter 10) to compare separation vectors at neighboring points and to define the "rate of change of separation." No transport, no comparison; no comparison, no meaning to the term "rate of change"! The notion of parallel transport founds itself

Tangent vector

Geodesic

Covariant derivative

Geodesic deviation

Spacetime curvature

This chapter: a Track-1 overview of differential geometry
on the idea of "geodesic," the world line of a freely falling particle. The special mathematical properties of a geodesic are explored in Chapter 10. That chapter uses geodesics to define parallel transport, uses parallel transport to define covariant derivative, and-completing the circle-uses covariant derivative to describe geodesics.

Chapter 11 faces up to the central issue: geodesic deviation ("rate of change of separation vector between two geodesics"), and its use in defining the curvature of spacetime.

But to define curvature is not enough. The man who would understand gravity deeply must also see curvature at work, producing relative accelerations of particles in Newtonian spacetime (Chapter 12); he must learn how, in Einstein spacetime, distances (metric) determine completely the curvature and the law of parallel transport (Chapter 13); he must be the master of powerful tools for computing curvature (Chapter 14); and he must grasp the geometric significance of the algebraic and differential symmetries of curvature (Chapter 15).

Unfortunately, such deep understanding requires time-far more time than one can afford in a ten-week or fifteen-week course, far more than a lone reader may wish to spend on first passage through the book. For the man who must rush on rapidly, this chapter contains a "Track-1" overview of the essential mathematical tools ( $\mathrm{\S }88.4-8.7$$\mathrm{\S }88.4-8.7$§88.4-8.7\S 88.4-8.7$\mathrm{\S }88.4-8.7$ ). From it one can gain an adequate, but not deep, understanding of spacetime curvature, of tidal gravitational forces, and of the mathematics of curved spacetime. This overview is also intended for the Track-2 reader; it will give him a taste of what is to come. The ambitious reader may also wish to consult other introductions to differential geometry (see Box 8.1).

There are several mathematics texts that may be consulted for a more detailed and extensive discussion of modern differential geometry along the line taken here. Bishop and Goldberg (1968) is the no. 1 reference. Hicks (1965) could be chosen as a current standard graduate-level text, with O'Neill (1966) at the undergraduate level introducing many of the same topics without presuming that the reader finds easy and obvious the current style in which pure mathematicians think and write. Auslander and MacKenzie (1963) at a somewhat more advanced level also allow for the reader to whom differential equations are more
familiar than homomorphisms. Willmore (1959) is easy to read but presents no challenge, and leads to little progress in adapting to the style of current mathematics. Trautman (1965) and Misner (1964a, 1969a) are introductions somewhat similar to ours, except for deemphasis of pictures; like ours, they are aimed at the student of relativity. Flanders (1963) is easy and useful as an introduction to exterior differential forms; it also gives examples of their application to a wide variety of topics in physics and engineering.

Nothing is more wonderful about the relation between Einstein's theory of gravity and Newton's theory than this, as discovered by Élie Cartan $(1923,1924)$$(1923,1924)$(1923,1924)(1923,1924)$(1923,1924)$ : that both theories lend themselves to description in terms of curvature; that in both this curvature is governed by the density of mass-energy; and that this curvature allows itself to be defined and measured without any use of or reference to any concept of metric. The difference between the two theories shows itself up in this: Einstein's theory in the end (or in the beginning, depending upon how one presents it!) does define an interval between every event and every nearby event; Newton's theory not only does not, but even says that any attempt to talk of spacetime intervals violates Newton's laws. This being the case, Track 2 will forego for a time (Chapters 9-12) any use of a spacetime metric ("Einstein interval"). It will extract everything possible from a metric-free description of spacetime curvature (all of Newton's theory; important parts of Einstein's theory).

Geodesic deviation is a measurer and definer of curvature, but the onlooker is forbidden to reduce a vector description of separation to a numerical measure of distance (no metric at this stage of the analysis): what an impossible situation! Nevertheless, that is exactly the situation with which Chapters $9-12$$9-12$9-129-12$9-12$ will concern themselves: how to do geometry without a metric. Speaking physically, one will overlook at this stage the fact that the geometry of the physical world is always and everywhere locally Lorentz, and endowed with a light cone, but one will exploit to the fullest the Galileo-Einstein principle of equivalence: in any given locality one can find a frame of reference in which every neutral test particle, whatever its velocity, is free of acceleration. The tracks of these neutral test particles define the geodesics of the geometry. These geodesics provide tools with which one can do much: define parallel transport (Chapter 10), define covariant derivative (Chapter 10), quantify geodesic deviation (Chapter 11), define spacetime curvature (Chapter 11), and explore Newtonian gravity (Chapter 12). Only after this full exploitation of metric-free geodesics will Track 2 admit the Einstein metric back into the scene (Chapters 13-15).

But to forego use of the metric is a luxury which Track 1 can ill afford; too little time would be left for relativistic stars, cosmology, black holes, gravitational waves, experimental tests, and the dynamics of geometry. Therefore, the Track-1 overview in this chapter keeps the Einstein metric throughout. But in doing so, it pays a heavy price: (1) no possibility of seeing curvature at work in Newtonian spacetime (Chapter 12); (2) no possibility of comparing and contrasting the geometric structures of Newtonian spacetime (Chapter 12) and Einstein spacetime (Chapter 13), and hence no possibility of grasping fully the Newtonian-based motivation for the Einstein field equations (Chapter 17); (3) no possibility of understanding fully the mathematical interrelationships of "geodesic," "parallel transport," "covariant derivative," "curvature," and "metric" (Chapters 9, 10, 11, 13); (4) no possibility of introducing the mathematical subjects "differential topology" (geometry without metric or covariant

Preview of Track-2 differential geometry

What the Track-1 reader will miss
derivative, Chapter 9) and "affine geometry" (geometry with covariant derivative but no metric, Chapters 10 and 11), subjects which find major application in modern analytical mechanics [see, e.g., Arnold and Avez (1968); also exercise 4.11 of this book], in Lie group theory with its deep implications for elementary particle physics [see, e.g., Hermann (1966); also exercises 9.12, 9.13, 10.16, and 11.12 of this book], in the theory and solution of partial differential equations [see, e.g., Sternberg (1969)], and, of course, in gravitation theory.

Gain the power in $\mathrm{\S }8.4$$\mathrm{\S }8.4$§8.4\S 8.4$\mathrm{\S }8.4$ and Chapter 9 to discuss tangent vectors, 1 -forms, tensors in curved spacetime; gain the power in $\mathrm{\S }8.5$$\mathrm{\S }8.5$§8.5\S 8.5$\mathrm{\S }8.5$ and Chapter 10 to parallel-transport vectors, to differentiate them, to discuss geodesics; use this power in $\mathrm{\$}8.7$$\mathrm{\$}8.7$$8.7\$ 8.7$\mathrm{\$}8.7$ and Chapter 11 to discuss geodesic deviation, to define curvature; .... But full power this will be only if it can be exercised in three ways: in pictures, in abstract notation, and in component notation (Box 8.3). Élie Cartan (Box 8.2) gave new insight into both

Geometry from three viewpoints: pictorial, abstract, component

Box 8.2 ÉLIE CARTAN, 1869-1951

Élie Cartan is a most remarkable figure in recent mathematical history. One learns from his obituary [Chern and Chevalley (1952)] that he was born a blacksmith's son in southern France, and proved the value of government scholarship aid by rising through the system to a professorship at the Sorbonne in 1912 when he was 43 . At the age of 32
he invented the exterior derivative [Cartan (1901)], which he used then mostly in differential equations and the theory of Lie groups, where he had already made significant contributions. He was about fifty when he began applying it to geometry, and sixty before Riemannian geometry specifically was the object of this research, including his text [Cartan (1928)], which is still reprinted and worth studying. Although universally recognized, his work did not find responsive readers until he neared retirement around 1940, when the "Bourbaki" generation of French mathematicians began to provide a conceptual framework for (among other things) Cartan's insights and methods. This made Cartan communicable and teachable as his own writings never were, so that by the time of his death at 82 in 1951 his influence was obviously dominating the revolutions then in full swing in all the fields (Lie groups, differential equations, and differential geometry) in which he had primarily worked.

The modern, abstract, coordinate-free approach to geometry, which is used extensively in this book, is due largely to Élie Cartan. He also discovered the geometric approach to Newtonian gravity that is developed and exploited in Chapter 12.

(1) Purely pictorial treatment of geometry:
tangent vector is conceived in terms of the separation of two points in the limit in which the points are indefinitely close;
vectors are added and subtracted locally as in flat space;
vectors at distinct points are compared by parallel transport from one point to another;
this parallel transport is accomplished by a "Schild's ladder construction" of geodesics (Box 10.2);
diagrams, yes; algebra, no;
it is tied conceptually as closely as possible to the world of test particles and measurements.
(2) Abstract differential geometry:
treats a tangent vector as existing in its own right, without necessity to give its breakdown into components,

just as one is accustomed nowadays in electromagnetism to treat the electric vector $\mathit{E}$$\mathit{E}$E\boldsymbol{E}$\mathit{E}$, without having to write out its components; uses a similar approach to differentiation (compare gradient operator $\mathbf{\nabla }$$\mathbf{\nabla }$grad\boldsymbol{\nabla}$\mathbf{\nabla }$ of elementary vector analysis, as distinguished from coor-dinate-dependent pieces of such an operator, such as $\partial /\partial x,\partial /\partial y$$\partial /\partial x,\partial /\partial y$del//del x,del//del y\partial / \partial x, \partial / \partial y$\partial /\partial x,\partial /\partial y$, etc.);
is the quickest, simplest mathematical scheme one knows to derive general results in differential geometry.
(3) Differential geometry as expressed in the language of components:
is indispensible in programming large parts of general relativity for a computer;
is convenient or necessary or both when one is dealing even at the level of elementary algebra with the most simple applications of relativity, from the expansion of the Friedmann universe to the curvature around a static center of attraction.

Newtonian gravity (Chapter 12) and the central geometric simplicity of Einstein's field equations (Chapter 15), because he had full command of all three methods of doing differential geometry. Today, no one has full power to communicate with others about the subject who cannot express himself in all three languages. Hence the interplay between the three forms of expression in what follows.

It is not new to go back and forth between the three languages, as witnesses the textbook treatment of the velocity and acceleration of a planet in Kepler motion around the sun. The velocity is written

(The hats " " on ${\mathit{e}}_{\hat{r}}$${\mathit{e}}_{\widehat{r}}$e_( hat(r))\boldsymbol{e}_{\hat{r}}${\mathit{e}}_{\hat{r}}$ and ${\mathit{e}}_{\hat{\varphi }}$${\mathit{e}}_{\widehat{\varphi }}$e_( hat(phi))\boldsymbol{e}_{\hat{\phi}}${\mathit{e}}_{\hat{\varphi }}$ signify that these are unit vectors.) The acceleration is

Planetary orbit as example of three viewpoints

Figure 8.1.
A Keplerian orbit in the sun's gravitational field, as treated using the standard Euclidean-space version of Newtonian gravity. The basis vectors themselves change from point to point along the orbit [equations (8.3)]. This figure illustrates the pictorial aspect of differential geometry. Later (exercise 8.5) it will illustrate the concepts of "covariant derivative" and "connection coefficients."

The unit vectors are turning (Figure 8.1) with the angular velocity $\omega =d\varphi /dt$$\omega =d\varphi /dt$omega=d phi//dt\omega=d \phi / d t$\omega =d\varphi /dt$; so

Thus the components of the acceleration have the values

and

Here is the acceleration in the language of components; $\mathit{a}$$\mathit{a}$a\boldsymbol{a}$\mathit{a}$ was the acceleration in abstract language; and Figure 8.1 shows the acceleration as an arrow. Each of these three languages will receive its natural generalization in the coming sections and chapters from two-dimensional flat space (with curvilinear coordinates) to four-dimensional curved spacetime, and from spacetime to more general manifolds (see $\mathrm{\S }9.7$$\mathrm{\S }9.7$§9.7\S 9.7$\mathrm{\S }9.7$ on manifolds).

Turn now to the Track-1 overview of differential geometry.

To see spacetime curvature at work, examine tidal gravitational forces (geodesic deviation); and to probe these forces, make measurements in a finite-sized laboratory. Squeeze the laboratory to infinitesimal size; all effects of spacetime curvature become infinitesimal; the physicist cannot tell whether he is in flat spacetime or curved spacetime. Neither can the mathematician, in the limit as his domain of attention squeezes down to a single event, ${\mathcal{P}}_0$${\mathcal{P}}_0$P_(0)\mathscr{P}_{0}${\mathcal{P}}_0$.

At the event ${\mathcal{P}}_o$${\mathcal{P}}_o$P_(o)\mathscr{P}_{o}${\mathcal{P}}_o$ (in the infinitesimal laboratory) both physicist and mathematician can talk of vectors, of 1 -forms, of tensors; and no amount of spacetime curvature can force the discussion to change from its flat-space form. A particle at ${\mathcal{P}}_o$${\mathcal{P}}_o$P_(o)\mathscr{P}_{o}${\mathcal{P}}_o$ has a 4 -momentum $\mathit{p}$$\mathit{p}$p\boldsymbol{p}$\mathit{p}$, with squared length

The squared length, as always, is calculated by inserting $\mathit{p}$$\mathit{p}$p\boldsymbol{p}$\mathit{p}$ into both slots of a linear machine, the metric $\mathit{g}$$\mathit{g}$g\boldsymbol{g}$\mathit{g}$ at ${\mathcal{P}}_0$${\mathcal{P}}_0$P_(0)\mathscr{P}_{0}${\mathcal{P}}_0$. The particle also has a 4-acceleration at ${\mathcal{P}}_o$${\mathcal{P}}_o$P_(o)\mathscr{P}_{o}${\mathcal{P}}_o$; and, if the particle is charged and freely moving, then $\mathit{a}$$\mathit{a}$a\boldsymbol{a}$\mathit{a}$ is produced by the electromagnetic field tensor $\mathit{F}$$\mathit{F}$F\boldsymbol{F}$\mathit{F}$ :

In no way can curvature affect such local, coordinate-free, geometric relations. And in no way can it prevent one from introducing a local Lorentz frame at ${\mathcal{P}}_o$${\mathcal{P}}_o$P_(o)\mathscr{P}_{o}${\mathcal{P}}_o$, and from performing standard, flat-space index manipulations in it:

But local Lorentz frames are not enough for the man who would calculate in curved spacetime. Non-Lorentz frames (nonorthonormal basis vectors $\left\{{\mathit{e}}_{\alpha }\right\}$$\left\{{\mathit{e}}_{\alpha }\right\}${e_(alpha)}\left\{\boldsymbol{e}_{\alpha}\right\}$\left\{{\mathit{e}}_{\alpha }\right\}$ ) often simplify calculations. Fortunately, no effort at all is required to master the rules of "index mechanics" in an arbitrary basis at a fixed event ${\mathcal{P}}_0$${\mathcal{P}}_0$P_(0)\mathscr{P}_{0}${\mathcal{P}}_0$. The rules are identical to those in flat spacetime, except that (1) the covariant Lorentz components ${\eta }_{\alpha \beta }$${\eta }_{\alpha \beta }$eta_(alpha beta)\eta_{\alpha \beta}${\eta }_{\alpha \beta }$ of the metric are replaced by

(2) the contravariant components ${\eta }^{\alpha \beta }$${\eta }^{\alpha \beta }$eta^(alpha beta)\eta^{\alpha \beta}${\eta }^{\alpha \beta }$ are replaced by $g^{\alpha \beta }$$g^{\alpha \beta }$g^(alpha beta)g^{\alpha \beta}$g^{\alpha \beta }$, where

i.e.,

(3) the Lorentz transformation matrix $\Vert {\mathrm{\Lambda }}^{\alpha }{}_{\beta }\Vert$$‖{\mathrm{\Lambda }}^{\alpha }{}_{\beta }‖$||Lambda^(alpha)_(beta)||\left\|\Lambda^{\alpha}{ }_{\beta}\right\|$\Vert {\mathrm{\Lambda }}^{\alpha }{}_{\beta }\Vert$ and its inverse $\Vert {\mathrm{\Lambda }}^{\beta }{}_{\alpha }\Vert$$‖{\mathrm{\Lambda }}^{\beta }{}_{\alpha }‖$||Lambda^(beta)_(alpha)||\left\|\Lambda^{\beta}{ }_{\alpha}\right\|$\Vert {\mathrm{\Lambda }}^{\beta }{}_{\alpha }\Vert$ are replaced by an arbitrary but nonsingular transformation matrix $\Vert L^{{\alpha }^{\prime }}{}_{\beta }\Vert$$‖L^{{\alpha }^{\prime }}{}_{\beta }‖$||L^(alpha^('))_(beta)||\left\|L^{\alpha^{\prime}}{ }_{\beta}\right\|$\Vert L^{{\alpha }^{\prime }}{}_{\beta }\Vert$ and its inverse $\Vert L^{\beta }{}_{\alpha }\Vert$$‖L^{\beta }{}_{\alpha }‖$||L^(beta)_(alpha)||\left\|L^{\beta}{ }_{\alpha}\right\|$\Vert L^{\beta }{}_{\alpha }\Vert$ :

Tensor algebra:
(1) occurs in infinitesimal neighborhood of an event
(2) is same in curved spacetime as in flat
(3) rules for component manipulation change slightly when using nonorthonormal basis
Components of metric

Transformation of basis
(4) in the special case of "coordinate bases," ${\mathit{e}}_{\alpha }=\partial \mathcal{P}/\partial x^{\alpha },{\mathit{e}}_{{\beta }^{\prime }}=\partial \mathcal{P}/\partial x^{{\beta }^{\prime }}$${\mathit{e}}_{\alpha }=\partial \mathcal{P}/\partial x^{\alpha },{\mathit{e}}_{{\beta }^{\prime }}=\partial \mathcal{P}/\partial x^{{\beta }^{\prime }}$e_(alpha)=delP//delx^(alpha),e_(beta^('))=delP//delx^(beta^('))\boldsymbol{e}_{\alpha}=\partial \mathscr{P} / \partial x^{\alpha}, \boldsymbol{e}_{\beta^{\prime}}=\partial \mathscr{P} / \partial x^{\beta^{\prime}}${\mathit{e}}_{\alpha }=\partial \mathcal{P}/\partial x^{\alpha },{\mathit{e}}_{{\beta }^{\prime }}=\partial \mathcal{P}/\partial x^{{\beta }^{\prime }}$,

Components of Levi-Civita tensor
and (5) the Levi-Civita tensor $\mathit{\epsilon }$$\mathit{\epsilon }$epsi\boldsymbol{\varepsilon}$\mathit{\epsilon }$, like the metric tensor, has components that depend on how nonorthonormal the basis vectors are (see exercise 8.3): if ${\mathit{e}}_0$${\mathit{e}}_0$e_(0)\boldsymbol{e}_{0}${\mathit{e}}_0$ points toward the future and ${\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3$${\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3$e_(1),e_(2),e_(3)\boldsymbol{e}_{1}, \boldsymbol{e}_{2}, \boldsymbol{e}_{3}${\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3$ are righthanded, then

where $[\alpha \beta \gamma \delta ]$$[\alpha \beta \gamma \delta ]$[alpha beta gamma delta][\alpha \beta \gamma \delta]$[\alpha \beta \gamma \delta ]$ is the completely antisymmetric symbol

and where $g$$g$gg$g$ is the determinant of the matrix $\Vert g_{\alpha \beta }\Vert$$‖g_{\alpha \beta }‖$||g_(alpha beta)||\left\|g_{\alpha \beta}\right\|$\Vert g_{\alpha \beta }\Vert$

Read Box 8.4 for full discussion and proofs; work exercise 8.1 below for fuller understanding and mastery.
Several dangers are glossed over in this discussion. In flat spacetime one often does not bother to say where a vector, 1 -form, or tensor is located. One freely moves geometric objects from event to event without even thinking. Of course, the unwritten rule of transport is: hold all lengths and directions fixed while moving; i.e., hold all Lorentz-frame components fixed; i.e., "parallel-transport" the object. But in

Tangent-vector basis: Pick ${\mathit{e}}_0,{\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3$${\mathit{e}}_0,{\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3$e_(0),e_(1),e_(2),e_(3)\boldsymbol{e}_{0}, \boldsymbol{e}_{1}, \boldsymbol{e}_{2}, \boldsymbol{e}_{3}${\mathit{e}}_0,{\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3$ at ${\mathcal{P}}_0$${\mathcal{P}}_0$P_(0)\mathscr{P}_{0}${\mathcal{P}}_0$ arbitrarily-but insist they be linearly independent.
"Dual basis" for 1-forms: The basis $\left\{{\mathit{e}}_{\alpha }\right\}$$\left\{{\mathit{e}}_{\alpha }\right\}${e_(alpha)}\left\{\boldsymbol{e}_{\alpha}\right\}$\left\{{\mathit{e}}_{\alpha }\right\}$ determines a 1-form basis $\left\{{\mathit{\omega }}^{\alpha }\right\}$$\left\{{\mathit{\omega }}^{\alpha }\right\}${omega^(alpha)}\left\{\boldsymbol{\omega}^{\alpha}\right\}$\left\{{\mathit{\omega }}^{\alpha }\right\}$ (its "dual basis") by

[see equation (2.19)].
Geometric interpretation (Figure 9.2): ${\mathit{e}}_2,{\mathit{e}}_3,{\mathit{e}}_0$${\mathit{e}}_2,{\mathit{e}}_3,{\mathit{e}}_0$e_(2),e_(3),e_(0)\boldsymbol{e}_{2}, \boldsymbol{e}_{3}, \boldsymbol{e}_{0}${\mathit{e}}_2,{\mathit{e}}_3,{\mathit{e}}_0$ lie parallel to surfaces of ${\mathit{\omega }}^1$${\mathit{\omega }}^1$omega^(1)\boldsymbol{\omega}^{1}${\mathit{\omega }}^1$; ${\mathit{e}}_1$${\mathit{e}}_1$e_(1)\boldsymbol{e}_{1}${\mathit{e}}_1$ pierces precisely one surface of ${\mathit{\omega }}^1$${\mathit{\omega }}^1$omega^(1)\boldsymbol{\omega}^{1}${\mathit{\omega }}^1$.
Function interpretation: $⟨{\mathit{\omega }}^{\alpha },{\mathit{e}}_{\beta }⟩={\delta }^{\alpha }{}_{\beta }$$⟨{\mathit{\omega }}^{\alpha },{\mathit{e}}_{\beta }⟩={\delta }^{\alpha }{}_{\beta }$(:omega^(alpha),e_(beta):)=delta^(alpha)_(beta)\left\langle\boldsymbol{\omega}^{\alpha}, \boldsymbol{e}_{\beta}\right\rangle=\delta^{\alpha}{ }_{\beta}$⟨{\mathit{\omega }}^{\alpha },{\mathit{e}}_{\beta }⟩={\delta }^{\alpha }{}_{\beta }$ determines the value of ${\mathit{\omega }}^{\alpha }$${\mathit{\omega }}^{\alpha }$omega^(alpha)\boldsymbol{\omega}^{\alpha}${\mathit{\omega }}^{\alpha }$ on any vector $\mathit{u}=u^{\beta }{\mathit{e}}_{\beta }$$\mathit{u}=u^{\beta }{\mathit{e}}_{\beta }$u=u^(beta)e_(beta)\boldsymbol{u}=u^{\beta} \boldsymbol{e}_{\beta}$\mathit{u}=u^{\beta }{\mathit{e}}_{\beta }$ (number of "bongs of bell" as $\mathit{u}$$\mathit{u}$u\boldsymbol{u}$\mathit{u}$ pierces ${\mathit{\omega }}^{\alpha }$${\mathit{\omega }}^{\alpha }$omega^(alpha)\boldsymbol{\omega}^{\alpha}${\mathit{\omega }}^{\alpha }$ ):

Special case: coordinate bases. Choose an arbitrary coordinate system $\{x^{\alpha }(\mathcal{P})\}$$\left\{x^{\alpha }(\mathcal{P})\right\}${x^(alpha)(P)}\left\{x^{\alpha}(\mathscr{P})\right\}$\{x^{\alpha }(\mathcal{P})\}$. At ${\mathcal{P}}_0$${\mathcal{P}}_0$P_(0)\mathscr{P}_{0}${\mathcal{P}}_0$ choose ${\mathit{e}}_{\alpha }=\partial \mathcal{P}/\partial x^{\alpha }$${\mathit{e}}_{\alpha }=\partial \mathcal{P}/\partial x^{\alpha }$e_(alpha)=delP//delx^(alpha)\boldsymbol{e}_{\alpha}=\partial \mathscr{P} / \partial x^{\alpha}${\mathit{e}}_{\alpha }=\partial \mathcal{P}/\partial x^{\alpha }$ as basis vectors. Then the dual basis is ${\mathit{\omega }}^{\alpha }=\mathit{d}{x}^{\alpha }$${\mathit{\omega }}^{\alpha }=\mathit{d}{x}^{\alpha }$omega^(alpha)=dx^(alpha)\boldsymbol{\omega}^{\alpha}=\boldsymbol{d} x^{\alpha}${\mathit{\omega }}^{\alpha }=\mathit{d}{x}^{\alpha }$. Proof: the general coordinate-free relation $⟨\mathit{d}f,\mathit{v}⟩={\partial }_{\mathit{v}}f$$⟨\mathit{d}f,\mathit{v}⟩={\partial }_{\mathit{v}}f$(:df,v:)=del_(v)f\langle\boldsymbol{d} f, \boldsymbol{v}\rangle=\partial_{\boldsymbol{v}} f$⟨\mathit{d}f,\mathit{v}⟩={\partial }_{\mathit{v}}f$ [equation (2.17)], with $f=x^{\alpha }$$f=x^{\alpha }$f=x^(alpha)f=x^{\alpha}$f=x^{\alpha }$ and $\mathit{v}=\partial \mathcal{P}/\partial x^{\beta }$$\mathit{v}=\partial \mathcal{P}/\partial x^{\beta }$v=delP//delx^(beta)\boldsymbol{v}=\partial \mathscr{P} / \partial x^{\beta}$\mathit{v}=\partial \mathcal{P}/\partial x^{\beta }$, reads

The Lorentz-frame discussion of equations (2.19) to (2.22) is completely unchanged when one switches to an arbitrary basis. Its conclusions:
expansion, $\mathit{u}={\mathit{e}}_{\alpha }{u}^{\alpha },\mathit{\sigma }={\sigma }_{\alpha }{\mathit{\omega }}^{\alpha }$$\mathit{u}={\mathit{e}}_{\alpha }{u}^{\alpha },\mathit{\sigma }={\sigma }_{\alpha }{\mathit{\omega }}^{\alpha }$u=e_(alpha)u^(alpha),sigma=sigma_(alpha)omega^(alpha)\boldsymbol{u}=\boldsymbol{e}_{\alpha} u^{\alpha}, \boldsymbol{\sigma}=\sigma_{\alpha} \boldsymbol{\omega}^{\alpha}$\mathit{u}={\mathit{e}}_{\alpha }{u}^{\alpha },\mathit{\sigma }={\sigma }_{\alpha }{\mathit{\omega }}^{\alpha }$;
calculation of components, $u^{\alpha }=⟨{\mathit{\omega }}^{\alpha },\mathit{u}⟩,{\sigma }_{\alpha }=⟨\mathit{\sigma },{\mathit{e}}_{\alpha }⟩$$u^{\alpha }=⟨{\mathit{\omega }}^{\alpha },\mathit{u}⟩,{\sigma }_{\alpha }=⟨\mathit{\sigma },{\mathit{e}}_{\alpha }⟩$u^(alpha)=(:omega^(alpha),u:),sigma_(alpha)=(:sigma,e_(alpha):)u^{\alpha}=\left\langle\boldsymbol{\omega}^{\alpha}, \boldsymbol{u}\right\rangle, \sigma_{\alpha}=\left\langle\boldsymbol{\sigma}, \boldsymbol{e}_{\alpha}\right\rangle$u^{\alpha }=⟨{\mathit{\omega }}^{\alpha },\mathit{u}⟩,{\sigma }_{\alpha }=⟨\mathit{\sigma },{\mathit{e}}_{\alpha }⟩$;
value of form on vector, $⟨\mathit{\sigma },\mathit{u}⟩={\sigma }_{\alpha }{u}^{\alpha }$$⟨\mathit{\sigma },\mathit{u}⟩={\sigma }_{\alpha }{u}^{\alpha }$(:sigma,u:)=sigma_(alpha)u^(alpha)\langle\boldsymbol{\sigma}, \boldsymbol{u}\rangle=\sigma_{\alpha} u^{\alpha}$⟨\mathit{\sigma },\mathit{u}⟩={\sigma }_{\alpha }{u}^{\alpha }$.
Application to gradients of functions:
expansion, $\mathit{d}f=f_{,\alpha }{\mathit{\omega }}^{\alpha }[$$\mathit{d}f=f_{,\alpha }{\mathit{\omega }}^{\alpha }\left[\right.$df=f_(,alpha)omega^(alpha)[:}\boldsymbol{d} f=f_{, \alpha} \boldsymbol{\omega}^{\alpha}\left[\right.$\mathit{d}f=f_{,\alpha }{\mathit{\omega }}^{\alpha }[$ defines $f_{,\alpha }]$$\left.f_{,\alpha }\right]${:f_(,alpha)]\left.f_{, \alpha}\right]$f_{,\alpha }]$;
calculation of components, $f_{,\alpha }=⟨\mathit{d}f,{\mathit{e}}_{\alpha }⟩={\partial }_{{\mathit{e}}_{\alpha }}f$$f_{,\alpha }=⟨\mathit{d}f,{\mathit{e}}_{\alpha }⟩={\partial }_{{\mathit{e}}_{\alpha }}f$f_(,alpha)=(:df,e_(alpha):)=del_(e_(alpha))ff_{, \alpha}=\left\langle\boldsymbol{d} f, \boldsymbol{e}_{\alpha}\right\rangle=\partial_{\boldsymbol{e}_{\alpha}} f$f_{,\alpha }=⟨\mathit{d}f,{\mathit{e}}_{\alpha }⟩={\partial }_{{\mathit{e}}_{\alpha }}f$ [see equation (2.17)].
Raising and lowering of indices is accomplished with $g_{\alpha \beta }$$g_{\alpha \beta }$g_(alpha beta)g_{\alpha \beta}$g_{\alpha \beta }$ and $g^{\alpha \beta }$$g^{\alpha \beta }$g^(alpha beta)g^{\alpha \beta}$g^{\alpha \beta }$ [equations (8.5) and (8.6)]. Proof:
$\tilde{\mathit{u}}$$\tilde{\mathit{u}}$widetilde(u)\widetilde{\boldsymbol{u}}$\tilde{\mathit{u}}$, the 1 -form corresponding to $\mathit{u}$$\mathit{u}$u\boldsymbol{u}$\mathit{u}$, is defined by $⟨\tilde{\mathit{u}},\mathit{v}⟩=\mathit{u}\cdot \mathit{v}$$⟨\tilde{\mathit{u}},\mathit{v}⟩=\mathit{u}\cdot \mathit{v}$(: widetilde(u),v:)=u*v\langle\widetilde{\boldsymbol{u}}, \boldsymbol{v}\rangle=\boldsymbol{u} \cdot \boldsymbol{v}$⟨\tilde{\mathit{u}},\mathit{v}⟩=\mathit{u}\cdot \mathit{v}$ for all $\mathit{v}$$\mathit{v}$v\boldsymbol{v}$\mathit{v}$; thus, $u_{\alpha }\equiv ⟨\tilde{\mathit{u}},{\mathit{e}}_{\alpha }⟩=\mathit{u}\cdot {\mathit{e}}_{\alpha }=u^{\beta }{\mathit{e}}_{\beta }\cdot {\mathit{e}}_{\alpha }=u^{\beta }{g}_{\beta \alpha };$$u_{\alpha }\equiv ⟨\tilde{\mathit{u}},{\mathit{e}}_{\alpha }⟩=\mathit{u}\cdot {\mathit{e}}_{\alpha }=u^{\beta }{\mathit{e}}_{\beta }\cdot {\mathit{e}}_{\alpha }=u^{\beta }{g}_{\beta \alpha };$u_(alpha)-=(:( widetilde(u)),e_(alpha):)=u*e_(alpha)=u^(beta)e_(beta)*e_(alpha)=u^(beta)g_(beta alpha);u_{\alpha} \equiv\left\langle\widetilde{\boldsymbol{u}}, \boldsymbol{e}_{\alpha}\right\rangle=\boldsymbol{u} \cdot \boldsymbol{e}_{\alpha}=u^{\beta} \boldsymbol{e}_{\beta} \cdot \boldsymbol{e}_{\alpha}=u^{\beta} g_{\beta \alpha} ;$u_{\alpha }\equiv ⟨\tilde{\mathit{u}},{\mathit{e}}_{\alpha }⟩=\mathit{u}\cdot {\mathit{e}}_{\alpha }=u^{\beta }{\mathit{e}}_{\beta }\cdot {\mathit{e}}_{\alpha }=u^{\beta }{g}_{\beta \alpha };$
inverting this equation yields $u^{\beta }=g^{\beta \alpha }{u}_{\alpha }$$u^{\beta }=g^{\beta \alpha }{u}_{\alpha }$u^(beta)=g^(beta alpha)u_(alpha)u^{\beta}=g^{\beta \alpha} u_{\alpha}$u^{\beta }=g^{\beta \alpha }{u}_{\alpha }$.

The discussion of Lorentz transformations in equations (2.39) to (2.43) is applicable to general changes of basis if one replaces $\Vert {\mathrm{\Lambda }}^{{\alpha }^{\prime }}{}_{\beta }\Vert$$‖{\mathrm{\Lambda }}^{{\alpha }^{\prime }}{}_{\beta }‖$||Lambda^(alpha^('))_(beta)||\left\|\Lambda^{\alpha^{\prime}}{ }_{\beta}\right\|$\Vert {\mathrm{\Lambda }}^{{\alpha }^{\prime }}{}_{\beta }\Vert$ by an arbitrary but nonsingular matrix $\Vert L^{{\alpha }^{\prime }}{}_{\beta }\Vert$$‖L^{{\alpha }^{\prime }}{}_{\beta }‖$||L^(alpha^('))_(beta)||\left\|L^{\alpha^{\prime}}{ }_{\beta}\right\|$\Vert L^{{\alpha }^{\prime }}{}_{\beta }\Vert$ [equations (8.7), (8.8)]. Conclusions:

When both bases are coordinate bases, then $L^{\beta }{}_{{\alpha }^{\prime }}=\partial x^{\beta }/\partial x^{{\alpha }^{\prime }},L^{{\alpha }^{\prime }}{}_{\beta }=\partial x^{{\alpha }^{\prime }}/\partial x^{\beta }$$L^{\beta }{}_{{\alpha }^{\prime }}=\partial x^{\beta }/\partial x^{{\alpha }^{\prime }},L^{{\alpha }^{\prime }}{}_{\beta }=\partial x^{{\alpha }^{\prime }}/\partial x^{\beta }$L^(beta)_(alpha^('))=delx^(beta)//delx^(alpha^(')),L^(alpha^('))_(beta)=delx^(alpha^('))//delx^(beta)L^{\beta}{ }_{\alpha^{\prime}}=\partial x^{\beta} / \partial x^{\alpha^{\prime}}, L^{\alpha^{\prime}}{ }_{\beta}=\partial x^{\alpha^{\prime}} / \partial x^{\beta}$L^{\beta }{}_{{\alpha }^{\prime }}=\partial x^{\beta }/\partial x^{{\alpha }^{\prime }},L^{{\alpha }^{\prime }}{}_{\beta }=\partial x^{{\alpha }^{\prime }}/\partial x^{\beta }$. Proof:

The discussions of tensor algebra given in $\mathrm{\S }3.2$$\mathrm{\S }3.2$§3.2\S 3.2$\mathrm{\S }3.2$ [equations (3.8) to (3.22)] and in $\mathrm{\S }3.5$$\mathrm{\S }3.5$§3.5\S 3.5$\mathrm{\S }3.5$ (excluding gradient and divergence) are unchanged, except that

and the components of the Levi-Civita tensor are changed from (3.50) to (8.10) [see exercise 8.3[.
Chief conclusions:
expansion, $\mathit{S}=S^{\alpha }{}_{\beta \gamma }{\mathit{e}}_{\alpha }\otimes {\mathit{\omega }}^{\beta }\otimes {\mathit{\omega }}^{\gamma };$$\mathit{S}=S^{\alpha }{}_{\beta \gamma }{\mathit{e}}_{\alpha }\otimes {\mathit{\omega }}^{\beta }\otimes {\mathit{\omega }}^{\gamma };$S=S^(alpha)_(beta gamma)e_(alpha)oxomega^(beta)oxomega^(gamma);\boldsymbol{S}=S^{\alpha}{ }_{\beta \gamma} \boldsymbol{e}_{\alpha} \otimes \boldsymbol{\omega}^{\beta} \otimes \boldsymbol{\omega}^{\gamma} ;$\mathit{S}=S^{\alpha }{}_{\beta \gamma }{\mathit{e}}_{\alpha }\otimes {\mathit{\omega }}^{\beta }\otimes {\mathit{\omega }}^{\gamma };$
components, $S^{\alpha }{}_{\beta \gamma }=\mathit{S}({\mathit{\omega }}^{\alpha },{\mathit{e}}_{\beta },{\mathit{e}}_{\gamma })$$S^{\alpha }{}_{\beta \gamma }=\mathit{S}\left({\mathit{\omega }}^{\alpha },{\mathit{e}}_{\beta },{\mathit{e}}_{\gamma }\right)$S^(alpha)_(beta gamma)=S(omega^(alpha),e_(beta),e_(gamma))S^{\alpha}{ }_{\beta \gamma}=\boldsymbol{S}\left(\boldsymbol{\omega}^{\alpha}, \boldsymbol{e}_{\beta}, \boldsymbol{e}_{\gamma}\right)$S^{\alpha }{}_{\beta \gamma }=\mathit{S}({\mathit{\omega }}^{\alpha },{\mathit{e}}_{\beta },{\mathit{e}}_{\gamma })$;
raising and lowering indices, $S_{\mu \beta }{}^{\nu }=g_{\mu \alpha }{g}^{\nu \gamma }{S}^{\alpha }{}_{\beta \gamma }$$S_{\mu \beta }{}^{\nu }=g_{\mu \alpha }{g}^{\nu \gamma }{S}^{\alpha }{}_{\beta \gamma }$S_(mu beta)^(nu)=g_(mu alpha)g^(nu gamma)S^(alpha)_(beta gamma)S_{\mu \beta}{ }^{\nu}=g_{\mu \alpha} g^{\nu \gamma} S^{\alpha}{ }_{\beta \gamma}$S_{\mu \beta }{}^{\nu }=g_{\mu \alpha }{g}^{\nu \gamma }{S}^{\alpha }{}_{\beta \gamma }$;
change of basis, $S^{{\lambda }^{\prime }}{}_{{\mu }^{\prime }{\nu }^{\prime }}=L^{{\lambda }^{\prime }}{}_{\alpha }{L}^{\beta }{}_{{\mu }^{\prime }}{L}^{\gamma }{}_{\nu }{}^{\prime }{S}^{\alpha }{}_{\beta \gamma }$$S^{{\lambda }^{\prime }}{}_{{\mu }^{\prime }{\nu }^{\prime }}=L^{{\lambda }^{\prime }}{}_{\alpha }{L}^{\beta }{}_{{\mu }^{\prime }}{L}^{\gamma }{}_{\nu }{}^{\prime }{S}^{\alpha }{}_{\beta \gamma }$S^(lambda^('))_(mu^(')nu^('))=L^(lambda^('))_(alpha)L^(beta)_(mu^('))L^(gamma)_(nu)^(')S^(alpha)_(beta gamma)S^{\lambda^{\prime}}{ }_{\mu^{\prime} \nu^{\prime}}=L^{\lambda^{\prime}}{ }_{\alpha} L^{\beta}{ }_{\mu^{\prime}} L^{\gamma}{ }_{\nu}{ }^{\prime} S^{\alpha}{ }_{\beta \gamma}$S^{{\lambda }^{\prime }}{}_{{\mu }^{\prime }{\nu }^{\prime }}=L^{{\lambda }^{\prime }}{}_{\alpha }{L}^{\beta }{}_{{\mu }^{\prime }}{L}^{\gamma }{}_{\nu }{}^{\prime }{S}^{\alpha }{}_{\beta \gamma }$;
machine operation, $\mathit{S}(\sigma ,\mathit{u},\mathit{v})=S^{\alpha }{}_{\beta \gamma }{}^{\sigma }{}_{\alpha }{u}^{\beta }{v}^{\gamma };$$\mathit{S}(\sigma ,\mathit{u},\mathit{v})=S^{\alpha }{}_{\beta \gamma }{}^{\sigma }{}_{\alpha }{u}^{\beta }{v}^{\gamma };$S(sigma,u,v)=S^(alpha)_(beta gamma)^(sigma)_(alpha)u^(beta)v^(gamma);\boldsymbol{S}(\sigma, \boldsymbol{u}, \boldsymbol{v})=S^{\alpha}{ }_{\beta \gamma}{ }^{\sigma}{ }_{\alpha} u^{\beta} v^{\gamma} ;$\mathit{S}(\sigma ,\mathit{u},\mathit{v})=S^{\alpha }{}_{\beta \gamma }{}^{\sigma }{}_{\alpha }{u}^{\beta }{v}^{\gamma };$
tensor product, $\mathit{T}=\mathit{u}\otimes \mathit{v}⟺T^{\alpha \beta }=u^{\alpha }{v}^{\beta };$$\mathit{T}=\mathit{u}\otimes \mathit{v}⟺T^{\alpha \beta }=u^{\alpha }{v}^{\beta };$T=u ox v LongleftrightarrowT^(alpha beta)=u^(alpha)v^(beta);\boldsymbol{T}=\boldsymbol{u} \otimes \boldsymbol{v} \Longleftrightarrow T^{\alpha \beta}=u^{\alpha} v^{\beta} ;$\mathit{T}=\mathit{u}\otimes \mathit{v}⟺T^{\alpha \beta }=u^{\alpha }{v}^{\beta };$
contraction, " $\mathit{M}=$$\mathit{M}=$M=\boldsymbol{M}=$\mathit{M}=$ contraction of $\mathit{R}$$\mathit{R}$R\boldsymbol{R}$\mathit{R}$ on slots 1 and $3"⟺M_{\mu \nu }=R^{\alpha }{}_{\mu \alpha \nu }$$3"⟺M_{\mu \nu }=R^{\alpha }{}_{\mu \alpha \nu }$3"LongleftrightarrowM_(mu nu)=R^(alpha)_(mu alpha nu)3 " \Longleftrightarrow M_{\mu \nu}=R^{\alpha}{ }_{\mu \alpha \nu}$3"⟺M_{\mu \nu }=R^{\alpha }{}_{\mu \alpha \nu }$;
wedge product, $\mathit{\alpha }\wedge \mathit{\beta }$$\mathit{\alpha }\wedge \mathit{\beta }$alpha^^beta\boldsymbol{\alpha} \wedge \boldsymbol{\beta}$\mathit{\alpha }\wedge \mathit{\beta }$ has components ${\alpha }^{\mu }{\beta }^{\nu }-{\beta }^{\mu }{\alpha }^{\nu }$${\alpha }^{\mu }{\beta }^{\nu }-{\beta }^{\mu }{\alpha }^{\nu }$alpha^(mu)beta^(nu)-beta^(mu)alpha^(nu)\alpha^{\mu} \beta^{\nu}-\beta^{\mu} \alpha^{\nu}${\alpha }^{\mu }{\beta }^{\nu }-{\beta }^{\mu }{\alpha }^{\nu }$;
Dual, ${}^{\ast }{J}_{\alpha \beta \gamma }=J^{\mu }{\epsilon }_{\mu \alpha \beta \gamma },{}^{\ast }{F}_{\alpha \beta }=\frac{1}{2}{F}^{\mu \nu }{\epsilon }_{\mu \nu \alpha \beta },{}^{\ast }{B}_{\alpha }=\frac{1}{6}{B}^{\lambda \mu \nu }{\epsilon }_{\lambda \mu \nu \alpha }$${}^{\ast }{J}_{\alpha \beta \gamma }=J^{\mu }{\epsilon }_{\mu \alpha \beta \gamma },{}^{\ast }{F}_{\alpha \beta }=\frac{1}{2}{F}^{\mu \nu }{\epsilon }_{\mu \nu \alpha \beta },{}^{\ast }{B}_{\alpha }=\frac{1}{6}{B}^{\lambda \mu \nu }{\epsilon }_{\lambda \mu \nu \alpha }$^(**)J_(alpha beta gamma)=J^(mu)epsi_(mu alpha beta gamma),^(**)F_(alpha beta)=(1)/(2)F^(mu nu)epsi_(mu nu alpha beta),^(**)B_(alpha)=(1)/(6)B^(lambda mu nu)epsi_(lambda mu nu alpha){ }^{*} J_{\alpha \beta \gamma}=J^{\mu} \varepsilon_{\mu \alpha \beta \gamma},{ }^{*} F_{\alpha \beta}=\frac{1}{2} F^{\mu \nu} \varepsilon_{\mu \nu \alpha \beta},{ }^{*} B_{\alpha}=\frac{1}{6} B^{\lambda \mu \nu} \varepsilon_{\lambda \mu \nu \alpha}${}^{\ast }{J}_{\alpha \beta \gamma }=J^{\mu }{\epsilon }_{\mu \alpha \beta \gamma },{}^{\ast }{F}_{\alpha \beta }=\frac{1}{2}{F}^{\mu \nu }{\epsilon }_{\mu \nu \alpha \beta },{}^{\ast }{B}_{\alpha }=\frac{1}{6}{B}^{\lambda \mu \nu }{\epsilon }_{\lambda \mu \nu \alpha }$.